I'm stuck right now on this equation: x(dy/dx) = 6y + (12x^4)[y^(2/3)]

The above equation looks nonlinear, and not separable. At the moment, I can't think of any method to employ when it comes to solving this equation for y, of the few methods I've learned so far up to this point. Again, separating variables didn't work for me, and it doesn't look like the integrating factor method will work either. Plus, it doesn't look exact at all.

This equation is linear. To put it into standard form, divide by x and rearrange:

du/dx - 2u/x = 4x^3

Your integrating factor appears to be correct. So other than a clerical mistake, you appear to have it nailed. Run your steps with the correction, then try putting the resulting solution back into the original equation to see if checks out.

This equation is linear. To put it into standard form, divide by x and rearrange:

du/dx - 2u/x = 4x^3

Your integrating factor appears to be correct. So other than a clerical mistake, you appear to have it nailed. Run your steps with the correction, then try putting the resulting solution back into the original equation to see if checks out.

I probably should've done the work on paper instead of typing it all out; maybe then, I could have avoided that careless error early on.

But yes, now I see my mistake. I'll continue from (du/dx) - (2/x)u = 4x^3: